Problem: All of the 3rd grade teachers and students from Covington went on a field trip to an art museum. Tickets were $$7.50$ each for teachers and $$4.00$ each for students, and the group paid $$55.00$ in total. A few weeks later, the same group visited a natural history museum where the tickets cost $$30.00$ each for teachers and $$7.50$ each for students, and the group paid $$135.00$ in total. Find the number of teachers and students on the field trips.
Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${7.5x+4y = 55}$ ${30x+7.5y = 135}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-30x-16y = -220}$ ${30x+7.5y = 135}$ Add the top and bottom equations together. $ -8.5y = -85 $ $ y = \dfrac{-85}{-8.5}$ ${y = 10}$ Now that you know ${y = 10}$ , plug it back into $ {7.5x+4y = 55}$ to find $x$ ${7.5x + 4}{(10)}{= 55}$ $7.5x+40 = 55$ $7.5x = 15$ $x = \dfrac{15}{7.5}$ ${x = 2}$ You can also plug ${y = 10}$ into $ {30x+7.5y = 135}$ and get the same answer for $x$ ${30x + 7.5}{(10)}{= 135}$ ${x = 2}$ There were $2$ teachers and $10$ students on the field trips.